old laptop won't turn on - help please

I have an old gateway laptop that I use where I do some volunteer work. It runs on Vista so you know it is an oldie. We keep it plugged in all the time because the battery won't hold a charge for long. Today I brought it home with me so I could work on my own time. Well, I didn't plug it back in for a couple hours and it must have died. So I plugged it in thinking it would recharge and I would be able to use it. I can't even get it to turn on now. I would think that being plugged in directly to electrical power I would be able to start it. When I push the power button a light at the front of the keyboard flashes red three times. What suggestion do you have as it has been working fine as long as it was plugged in. Do I need to remove the battery and see if it will start when plugged in. Not wanting to look for a replacement battery since it is so old and no guarantee that would get it running. I do have the file backed up on a flash drive. Without that computer I can't do any volunteer work for them. I don't think they want or need to buy a new computer as this was only used for some temporary volunteer work I was doing.
Appreciate any ideas to get it on again.

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Sometimes, removing the battery and power plug, then holding down the power button for 10 seconds will reset the laptop so that it will work again when you reconnect everything. Revived three different laptops this way.

The only other things besides pulling the battery and deep capacitor discharge above is to check is that there is actually AC power coming out of the socket. The red flashing LED may be from residual power available from the battery so does not rule out no power.

Make sure the adapter is fully seated and the connection is not loose.

And if you have a volt meter, that the wall adapter is putting out its designated voltage (not under load however, so a pass is not a pass, but a fail is a fail).

If the battery was already a problem (unable to hold a charge) and you have no other specific reason to suspect any other component (I'll assume you didn't drop it or pour a cup of coffee on it), then this is the rule. It is at least four-fifths likely that the battery is the problem here.

Batteries are a classic trouble spot. So are screens, ports, hinges and fans. If you can rule out screens, ports, hinges and fans as a probable cause... then you are left with the battery. Which you already know is causing trouble.

This leaves you (and the volunteer organization) with a problem. You can:

1). Replace the battery and hope that gets the laptop back up and running;
2). Campaign for a new laptop. You can be certain that will get you a working computer back again, but at the cost of more money;
3). Do nothing and live without a computer.